Elliptic curve cryptography is an approach to public key cryptography based on the algebraic structure of elliptic curves over a finite field, also referred to as Galois field. According to an elliptic curve based protocol it is assumed that finding the discrete logarithm of a random elliptic curve element with respect to a publicly known base point (a point on the elliptic curve) is unfeasible. Thereby, the size of the elliptic curve may determine the difficulty of the problem. An elliptic curve is a plane curve, which consists of points (x,y) satisfying the elliptic curve equation along with a distinguished point at infinity. This set of points on the curve and the point at infinity form together with the group operation (elliptic curve addition operation) an Abelian group with the point at infinity as identity element.
Herein, the elliptic curve is defined over a finite field such that the number of elements x over which the elliptic curve is defined, is finite. For the points on the elliptic curve given by the two components x and y, wherein x and y satisfy the elliptic curve equation, an elliptic curve addition operation is defined. Multiple application of the elliptic curve addition operation may define an elliptic curve multiplication operation, wherein the multiplication of a point is an integer multiplication of that point.
The publication “Use of Elliptic Curves in Cryptography”, Victor Miller, Crypto '85 discusses the use of elliptic curves in cryptography, wherein a key exchange protocol is proposed, which appears to be immune from a attacks of the style of Western, Miller and Adleman.
The publication “Using Elliptic Curves on RFID Tags”, by Braun, Hess, Meyer, International Journal of Computer Science and Network Security, Vol. 8, No. 2, February 2008 discloses a concept for the realization of asymmetric cryptographic techniques in light-weight cryptographic devices and describes an implementation based on elliptic curve cryptography, which can be used for authentication in mass applications of RFID-tags.